Analysis of Variance
Department of Educational Psychology
Agenda
1 Overview and Introduction
2 Basics of the Chi-Square Distribution
3 Goodness-of-Fit Test
4 Test of Independence
5 Test of Homogeneity
6 Conclusion
Agenda
1 Overview and Introduction
2 Basics of the Chi-Square Distribution
3 Goodness-of-Fit Test
4 Test of Independence
5 Test of Homogeneity
6 Conclusion
Agenda
1 Overview and Introduction
2 Basics of the Chi-Square Distribution
3 Goodness-of-Fit Test
4 Test of Independence
5 Test of Homogeneity
6 Conclusion
\[ \chi^2 = \sum_k\frac{(O - E)^2}{E} \]
\[ df = k - 1 \]
| Grade | Expected (E) | Observed (O) | (O − E) | ’’ / E |
|---|---|---|---|---|
| A | 4 | 2 | -2 | (−2)² / 4 = 1.00 |
| B | 4 | 4 | 0 | 0.00 |
| C | 4 | 12 | 8 | 8² / 4 = 16.00 |
| D | 4 | 1 | -3 | (−3)² / 4 = 2.25 |
| F | 4 | 1 | -3 | (−3)² / 4 = 2.25 |
| Total | 20 | 20 | — | 21.50 |
Agenda
1 Overview and Introduction
2 Basics of the Chi-Square Distribution
3 Goodness-of-Fit Test
4 Test of Independence
5 Test of Homogeneity
6 Conclusion
\[ \chi^2 = \sum_{i * j}\frac{(O - E)^2}{E} \]
\[ df = (i - 1) * (j - 1) \]
| Grade | Female (F) | Male (M) | Total |
|---|---|---|---|
| A | 6 | 3 | 9 |
| B | 5 | 4 | 9 |
| C | 3 | 4 | 7 |
| D | 5 | 7 | 12 |
| F | 1 | 2 | 3 |
| Total | 20 | 20 | 40 |
| Grade | Female (E) | Male (E) |
|---|---|---|
| A | 9×20/40 = 4.5 | 4.5 |
| B | 9×20/40 = 4.5 | 4.5 |
| C | 7×20/40 = 3.5 | 3.5 |
| D | 12×20/40 = 6.0 | 6.0 |
| F | 3×20/40 = 1.5 | 1.5 |
| Grade | Female (O−E)²/E | Male (O−E)²/E |
|---|---|---|
| A | (6−4.5)²/4.5 = 0.50 | (3−4.5)²/4.5 = 0.50 |
| B | (5−4.5)²/4.5 = 0.06 | (4−4.5)²/4.5 = 0.06 |
| C | (3−3.5)²/3.5 = 0.07 | (4−3.5)²/3.5 = 0.07 |
| D | (5−6.0)²/6.0 = 0.17 | (7−6.0)²/6.0 = 0.17 |
| F | (1−1.5)²/1.5 = 0.17 | (2−1.5)²/1.5 = 0.17 |
| Total χ² = 1.94 |
Agenda
1 Overview and Introduction
2 Basics of the Chi-Square Distribution
3 Goodness-of-Fit Test
4 Test of Independence
5 Test of Homogeneity
6 Conclusion
\[ \chi^2 = \sum_{i * j}\frac{(O - E)^2}{E} \]
\[ df = (i - 1) * (j - 1) \]
| Class | Female (F) | Male (M) | Total |
|---|---|---|---|
| Freshman | 6 | 5 | 11 |
| Sophomore | 14 | 7 | 21 |
| Junior | 8 | 7 | 15 |
| Senior | 12 | 8 | 20 |
| Total | 40 | 27 | 67 |
| Class | Expected F | Expected M |
|---|---|---|
| Freshman | 6.57 | 4.43 |
| Sophomore | 12.54 | 8.46 |
| Junior | 8.96 | 6.04 |
| Senior | 11.94 | 8.06 |
| Class | Group | (O) | (E) | ((O-E)^2/E) |
|---|---|---|---|---|
| Freshman | F | 6 | 6.57 | 0.049 |
| Freshman | M | 5 | 4.43 | 0.074 |
| Sophomore | F | 14 | 12.54 | 0.170 |
| Sophomore | M | 7 | 8.46 | 0.252 |
| Junior | F | 8 | 8.96 | 0.103 |
| Junior | M | 7 | 6.04 | 0.151 |
| Senior | F | 12 | 11.94 | 0.0003 |
| Senior | M | 8 | 8.06 | 0.0004 |
Agenda
1 Overview and Introduction
2 Basics of the Chi-Square Distribution
3 Goodness-of-Fit Test
4 Test of Independence
5 Test of Homogeneity
6 Conclusion
The chi-squared distribution is well suited to dealing with categorical counts and proportions, but the 3 forms of the chi-square distribution have somewhat different application and formulas
The shape of the chi-square distribution changes based upon the dfs, and the hypothesis testing done with it is usually right-tailed, among other consistent characteristics
Like with the other distributions we have discussed, be mindful of how the null and alternative hypotheses change
Module 11 Lecture - Review of Prior Topics || Analysis of Variance